Based on the above preliminary considerations we can now prove the main result of this
section, which connects the Maslov index with the Lidskii angles and hence with the
oscillation number through equation (REF ). More precisely, the
Maslov index of \(Y\)  and \(\hat{Y}\)  over the interval \([a,b]\)  can be calculated as the
total change in the interval \([a,b]\)  of the integers \(q_j(t)\) , which are associated
through (REF ) or (REF ) with the continuous Lidskii
angles \(\varphi _j(t)\)  of the symplectic matrix \(S(t)\)  defined in (REF ),
compare with [1]} and [2]}.
Consequently, the Maslov index is equal to the oscillation number of the transformed
Lagrangian path \(Z_Y^{-1}\hspace*{1.0pt}\hat{Y}\)  on \([a,b]\) .
